Suresh Govindarajan

We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a sl(2)ˆ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of sl(2)ˆ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.

Stationary, supersymmetric supergravity solutions in five dimensions have Kähler metrics on the four-manifold orthogonal to the orbits of a time-like Killing vector. We show that an explicit class of toric Kähler metrics provide a unified framework in which to describe both the asymptotically flat and asymptotically AdS solutions. The Darboux co-ordinates used for the local description turn out to be ''ring-like.'' We conclude with an ansatz for studying the existence of supersymmetric black rings in AdS. © SISSA 2007.

We compute the partition function for the topological Landau-Ginzburg B-model on the disk. This is done by treating the worldsheet superpotential perturbatively. We argue that this partition function as a function of bulk and boundary perturbations may be identified with the effective D-brane superpotential in the target spacetime. We point out the relationship of this approach to matrix factorizations. Using these methods, we prove a conjecture for the effective superpotential of Herbst, Lazaroiu and Lerche for the A-type minimal models. We also consider the Landau-Ginzburg theory of the cubic torus where we show that the effective superpotential, given by the partition function, is consistent with the one obtained by summing up disk instantons in the mirror A-model. This is done by explicitly constructing the open-string mirror map. © SISSA 2006.

In this paper we study the relation between parabolic Higgs vector bundles and irreducible representations of the fundamental group of punctured Riemann surfaces established by Simpson. We generalize a result of Hitchin, identifying those parabolic Higgs bundles that correspond to Fuchsian representations. We also study the Higgs bundles that give representations whose image is contained, after conjugation, in SL(fc,R). We compute the real dimension of one of the components of this space of representations, which in the absence of punctures is the generalized Teichmüller space introduced by Hitchin, and which in the case of k = 2 is the usual Teichmüller space of the punctured surface. ©1997 American Mathematical Society.

Recently, certain higher-dimensional complex manifolds were obtained by S. Govindarajan [1] by associating a higher dimensional uniformisation to the generalised Teichmüller spaces of Hitchin. The extra dimensions are provided by the 'times' of the generalised KdV hierarchy. In this Letter, we complete the proof that these manifolds provide the analog of superspace for W-gravity and that W-symmetry linearises on these spaces. This is done by explicitly constructing the relationship between the Beltrami differentials which naturally occur in the higher-dimensional manifolds and the Beltrami differentials which occur in W-gravity. This also resolves an old puzzle regarding the relationship between KdV flows and W-diffeomorphisms.

We present a method based on mutations of helices which leads to the construction (in the large-volume limit) of exceptional coherent sheaves associated with the (∑ala=0) orbits in Gepner models. This is explicitly verified for a few examples including some cases where the ambient weighted projective space has singularities not inherited by the Calabi-Yau hypersurface. The method is based on two conjectures which lead to the analog, in the general case, of the Beilinson quiver for Pn. We discuss how one recovers the McKay quiver using the gauged linear sigma model (GLSM) near the orbifold or Gepner point in Kähler moduli space.

The motion of a particle moving under the influence of a central force is a fundamental paradigm in dynamics. The problem of planetary motion, specifically the derivation of Kepler’s laws motivated Newton’s monumental work, Principia Mathematica, effectively signalling the start of modern physics. Today, the central force problem stands as a basic lesson in dynamics. In this article, we discuss the classical central force problem in a general number of spatial dimensions n, as an instructive illustration of important aspects such as integrability, super-integrability and dynamical symmetry. The investigation is also in line with the realisation that it is useful to treat the number of dimensions as a variable parameter in physical problems. The dependence of various quantities on the spatial dimensionality leads to a proper perspective of the problems concerned. We consider, first, the orbital angular momentum (AM) in n dimensions, and discuss in some detail the role it plays in the integrability of the central force problem. We then consider an important super-integrable case, the Kepler problem, in n dimensions. The existence of an additional vector constant of the motion (COM) over and above the AM makes this problem maximally super-integrable. We discuss the significance of these COMs as generators of the dynamical symmetry group of the Hamiltonian. This group, the rotation group in n + 1 dimensions, is larger than the kinematical symmetry group for a general central force, namely, the rotation group in n dimensions.

We study fractional 2p -branes and their intersection numbers in non-compact orbifolds as well the continuation of these objects in Kähler moduli space to coherent sheaves in the corresponding smooth non-compact Calabi-Yau manifolds. We show that the restriction of these objects to compact Calabi-Yau hypersurfaces gives the new fractional branes in LG orbifolds constructed by Ashok et. al. in [13]. We thus demonstrate the equivalence of the B-type branes corresponding to linear boundary conditions in LG orbifolds, originally constructed in [2], to a subset of those constructed in LG orbifolds using boundary fermions and matrix factorization of the world-sheet superpotential. The relationship between the coherent sheaves corresponding to the fractional two-branes leads to a generalization of the McKay correspondence that we call the quantum McKay correspondence due to a close parallel with the construction of branes on non-supersymmetric orbifolds. We also provide evidence that the boundary states associated to these branes in a conformal field theory description corresponds to a sub-class of the boundary states associated to the permutation branes in the Gepner model associated with the LG orbifold. © SISSA 2005.

We provide evidence for the existence of a family of generalized Kac-Moody (GKM) superalgebras, N whose Weyl-Kac-Borcherds denominator formula gives rise to a genus-two modular form at level N, Δ(Z), for (N,k) = (1,10), (2,6), (3,4), and possibly (5,2). The square of the automorphic form is the modular transform of the generating function of the degeneracy of CHL dyons in asymmetric N-orbifolds of the heterotic string compactified on T6. The new generalized Kac-Moody superalgebras all arise as different 'automorphic corrections' of the same Lie algebra and are closely related to a generalized Kac-Moody superalgebra constructed by Gritsenko and Nikulin. The automorphic forms, Δ2(Z), arise as additive lifts of Jacobi forms of (integral) weight k/2 and index 1/2. We note that the orbifolding acts on the imaginary simple roots of the unorbifolded GKM superalgebra, 1, leaving the real simple roots untouched. We anticipate that these superalgebras will play a role in understanding the 'algebra of BPS states' in CHL compactifications. © 2009 SISSA.

We systematically study and obtain the large-volume analogues of fractional two-branes on resolutions of the orbifolds ℂ3/ℤ n. We also study a generalisation of the McKay correspondence proposed in hep-th/0504164 called the quantum McKay correspondence by constructing duals to the fractional two-branes. Details are explicitly worked out for two examples - the crepant resolutions of ℂ3/ ℤ3 and ℂ3/ℤ5. © SISSA 2006.